Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra
Abstract
Let A be a finite dimensional algebra and let rmHH*(A) be its Hochschild cohomology ring, which is a Gerstenhaber algebra. Denote by calN (resp. G, calG) the ideal (resp. weak Gerstenhaber ideal, Gerstenhaber ideal) generated by all homogeneous nilpotent elements. Motivated by their work on support varieties via Hochschild cohomology, Snashall and Solberg conjectured that rmHH*(A)/calN is a finitely generated algebra. Xu constructed a counterexample to the Snashall-Solberg conjecture over a base field of characteristic two, and Snashall generalized this example to arbitrary characteristic. Hermann further asked whether rmHH*(A)/G is a finitely generated algebra and suggested considering first the Xu--Snashall algebra. In this paper, we answer this question for the Xu--Snashall algebra. In fact, by explicitly computing the Gerstenhaber algebra structure on the Hochschild cohomology ring, we show that G=calN; hence rmHH*(A)/G=rmHH*(A)/calN is not a finitely generated algebra. Furthermore, we show that rmHH*(A)/calG≅ K. Therefore, one may ask whether, for a finite dimensional algebra A, rmHH*(A)/calG is always a finitely generated algebra. Our main tools are two-sided Anick resolutions and weak self-homotopies.
Related
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BibTeX
@misc{long2026gerstenhaber,
title = {{Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra}},
author = {Qi Long and Ziyang Shi and Guodong Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19560},
eprint = {2609.19560},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19560},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra
AU - Qi Long
AU - Ziyang Shi
AU - Guodong Zhou
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19560
UR - https://arxiv.org/abs/2609.19560
AB - Let A be a finite dimensional algebra and let rmHH^*(A) be its Hochschild cohomology ring, which is a Gerstenhaber algebra. Denote by calN (resp. G, calG) the ideal (resp. weak Gerstenhaber ideal, Gerstenhaber ideal) generated by all homogeneous nilpotent elements. Motivated by their work on support varieties via Hochschild cohomology, Snashall and Solberg conjectured that rmHH^*(A)/calN is a finitely generated algebra. Xu constructed a counterexample to the Snashall-Solberg conjecture over a base field of characteristic two, and Snashall generalized this example to arbitrary characteristic. Hermann further asked whether rmHH^*(A)/G is a finitely generated algebra and suggested considering first the Xu--Snashall algebra. In this paper, we answer this question for the Xu--Snashall algebra. In fact, by explicitly computing the Gerstenhaber algebra structure on the Hochschild cohomology ring, we show that G=calN; hence rmHH^*(A)/G=rmHH^*(A)/calN is not a finitely generated algebra. Furthermore, we show that rmHH^*(A)/calG≅ K. Therefore, one may ask whether, for a finite dimensional algebra A, rmHH^*(A)/calG is always a finitely generated algebra. Our main tools are two-sided Anick resolutions and weak self-homotopies.
ER -